Generalized Hurwitz matrices, generalized Euclidean algorithm, and forbidden sectors of the complex plane
| Autor: | Holtz, Olga, Khrushchev, Sergey, Kushel, Olga |
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| Rok vydání: | 2015 |
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| Zdroj: | Comput. Methods Funct. Theory 16 (2016), no.3, 395 - 431 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1007/s40315-016-0156-0 |
| Popis: | Given a polynomial \[ f(x)=a_0x^n+a_1x^{n-1}+\cdots +a_n \] with positive coefficients $a_k$, and a positive integer $M\leq n$, we define a(n infinite) generalized Hurwitz matrix $H_M(f):=(a_{Mj-i})_{i,j}$. We prove that the polynomial $f(z)$ does not vanish in the sector $$ \left\{z\in\mathbb{C}: |\arg (z)| < \frac{\pi}{M}\right\} $$ whenever the matrix $H_M$ is totally nonnegative. This result generalizes the classical Hurwitz' Theorem on stable polynomials ($M=2$), the Aissen-Edrei-Schoenberg-Whitney theorem on polynomials with negative real roots ($M=1$), and the Cowling-Thron theorem ($M=n$). In this connection, we also develop a generalization of the classical Euclidean algorithm, of independent interest per se. Comment: 26 pages, 1 figure |
| Databáze: | arXiv |
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